6.2 Integral Applications
Question #5 (Hard): Finding Volume Equation for Geometric Solids
Sometimes geometric shapes are not round cylindrical shapes, but angular shapes such as pyramids,
frustum, tetrahedrons with perpendicular faces. No matter what shape it is, first the area function
expressing the cross section of the solid needs to be determined first. The height can be kept
along the -axis, unless the question specifies otherwise (eg. vertically rising, in which case the height is
along the -axis and integral over the variable). The interval then becomes [ ]. The linear ratio
relationship of the sides of the cross section to the given dimension of the solid is typically used to
establish the expression for . Rearranging the ratio, the can be written in terms of the
given dimensions and over which the integral is evaluated.
Find the volume of a frustum of a pyramid with equilateral triangle base with side length of and
equilateral top with a side length of and height .
To get the volume of the frustum, take the volume of the larger pyramid then subtract by the smaller
pyramid. First, work with general pyramid, not a frustum:
An equilateral triangle with side length of has its altitude expressed as:
√ ( ) √ ( ) √ √ . So the area